CN106763650A - The tooth form extracting method of involute helicoid worm - Google Patents

Abstract

The invention discloses a kind of tooth form extracting method of involute cylindrical worm, including coordinate system is set up, set up the parametric equation of flank profil, its parametric equation is carried out into spatial variations, set up the steps such as the parametric equation of complete tooth form.The present invention obtains the tooth form of involute cylindrical worm by the way of transverse profile develops along helix, the intersection of its tooth form and normal section is obtained into normal tooth profile of the involute cylindrical worm in horizontal plane by spatial alternation, avoid and the normal tooth profile of involute cylindrical worm is produced the problem of larger error as involute to process in many cases at present, while being also convenient for observing the shape of its normal tooth profile.

Description

The tooth form extracting method of involute helicoid worm

Technical field

Field is extracted the invention belongs to worm tooth shape, and in particular to the normal tooth profile to a kind of involute cylindrical worm is extracted Method.

Background technology

Involute cylindrical worm has bearing capacity strong, stable movement, transmission efficiency, the advantages of long lifespan, uses extensively In national defence and civilian industry field.The transverse profile of involute helicoid worm is involute, and it is equivalent to the few number of teeth, a large helix angle Divergent exhaust pipes, worm tooth-surface is involute helical tooth flank.When rotating is realized, it can overcome straight sided axial worm Asymmetric shortcoming is driven during transmission, and bearing capacity and transmission efficiency are higher.

The tooth form extraction accuracy of involute cylindrical worm affects the transmission performance of involute cylindrical worm.With parametrization Technology, designer only needs to In-put design principal parameter and can just drive the geometrical model of product parts.Existing extractive technique exists It is in many cases that its normal tooth profile is also gone into treatment as involute, and actually the normal tooth profile of involute cylindrical worm is simultaneously Do not define accurately, it is not involute, it is this approximately to be caused instead of the way of normal direction tooth curve with involute The tooth form of extraction is inaccurate.Also what is had has simply done some simple analyses, does not have for its normal tooth profile and is accurately given Its Mathematical Modeling and specific method for drafting, simply depicting substantially, can not be accurate and complete extract gradually is opened The normal tooth profile of line cylindrical worm.

The content of the invention

It is an object of the invention to provide a kind of tooth form extracting method of involute cylindrical worm, using known flank of tooth side Journey and space coordinate transformation, by by two helicoids of tooth form being that complete being derived of model converted again, from And extract the normal tooth profile of involute cylindrical worm.

The purpose of the present invention is achieved through the following technical solutions, the tooth form extracting method of involute cylindrical worm, bag Include following steps：

The first step：The parameter for determining involute cylindrical worm to be designed is:Axial module m, head number z₁, rotation direction, pressure angle α, centre-to-centre spacing a, lead angle γ, reference diameter d₁；

Second step：It is XY faces with the plane vertical with the axis of involute cylindrical worm to be designed, with involute to be designed The axis of cylindrical worm is Z axis, sets up OXYZ coordinate systems, wherein, tooth form is symmetrical on X-axis；By the property of involute, in OXYZ The equation of involute HG on the right side of being obtained in coordinate system：

3rd step：For bull involute cylindrical worm, tooth form right side involute HG need to be through such as down conversion：

4th step：Involute HG is spinned about the z axis to move and forms right side helicoid, formula is turned round according to vector, this is gradually Burst at the seams helicoid equation be (left-handed z takes negative sign, and dextrorotation takes positive sign)：

5th step：The normal tooth profile of involute cylindrical worm is extracted for convenience, can be by transverse profile in X-O-Y planes On about the z axis positive and negative both direction rotation, i.e. z₁₁=0.λ can then be obtained into portion according to design requirement in 0 neighbouring value Divide helicoid：

6th step：By the Mathematical Modeling obtained by previous step around X-axis rotate counterclockwiseThe involute cylinder for then obtaining The intersection of worm tooth-surface and horizontal plane is the Mathematical Modeling of its normal tooth profile：

Z=0-- horizontal planes, that is, the normal tooth profile chosen is the intersection of the involute helicoid worm flank of tooth and the plane where z=0

7th step：The equation of involute FE on the left of being obtained in OXYZ coordinate systems：

8th step：Similarly, the equation to the 7th step involute FE carries out the conversion of the 4th step to the 6th step again, can this gradually The Mathematical Modeling of the left-hand screw face normal tooth profile that dextrorotation about the z axis of bursting at the seams is formed：

x₂₂₂, y₂₂₂, z₂₂₂It is that n-th right side involute FE of multistart worm does the λ=0 part tooth formed after spatial alternation The coordinate of any one point on the involute FE in face；

9th step：Design parameter and the 6th step according to involute cylindrical worm to be designed, the normal direction that the 8th step is released Three equations of the flank of tooth of tooth form, the intersection in the involute cylindrical worm flank of tooth He its normal section obtains intersection point, is gradually to open The normal tooth profile of line cylindrical worm；

Compared with prior art, the beneficial effects of the invention are as follows：The tooth form extracting method of involute cylindrical worm, both avoided Approximately the tooth form extracted is caused to produce instead of the way of normal tooth profile using involute cylindrical worm end section involute larger Error, and the accurate and complete Mathematical Modeling for giving normal tooth profile and its figure.

Brief description of the drawings：

Fig. 1 be involute cylindrical worm of the present invention tooth form extracting method in the monodentate whole profile set up of second step Schematic diagram.

Fig. 2 be involute cylindrical worm of the present invention tooth form extracting method in the four-head worm screw full teeth set up of the 3rd step Wide schematic diagram.

Fig. 3 be involute cylindrical worm of the present invention tooth form extracting method in the four-head dextrorotation involute set up of the 5th step The tooth form schematic diagram of cylindrical worm.

Fig. 4 is the part set up of the step of tooth form extracting method the 6th and the 8th step of involute cylindrical worm of the present invention through sky Between convert after helicoid and horizontal plane, its intersection is exactly the schematic diagram of normal tooth profile.

Fig. 5 is that the normal tooth profile set up of the step of tooth form extracting method the 9th of involute cylindrical worm of the present invention is illustrated Figure.

Specific embodiment

With reference to specific embodiment, the present invention is described in detail.

The tooth form extracting method of involute cylindrical worm, comprises the following steps：

The first step, by taking set below parameter as an example, as the known parameters of the tooth form of involute cylindrical worm to be extracted：Axle To modulus m=2.563mm, head number z₁=4 (dextrorotation), pressure angle α=20 °, centre-to-centre spacing a=50mm, lead angle γ= 23.905 °, reference diameter d₁=23.125mm.

Second step：It is XY faces with the plane vertical with the axis of involute cylindrical worm to be designed, with involute to be designed The axis of cylindrical worm is Z axis, sets up OXYZ coordinate systems, wherein, tooth form is symmetrical on X-axis, by the property of involute, in OXYZ The equation of involute HG on the right side of being obtained in coordinate system, as shown in Figure 1；

3rd step：For four-head involute cylindrical worm, tooth form right side involute HG need to be through such as down conversion：

4th step：Involute HG is spinned about the z axis to move and forms right side helicoid, formula is turned round according to vector, this is gradually The equation of helicoid of bursting at the seams is：

5th step：The normal tooth profile of involute cylindrical worm is extracted for convenience, can be by transverse profile in X-O-Y planes On about the z axis positive and negative both direction rotation, i.e. z₁₁=0.λ is taken into λ in 0 neighbouring value according to design requirement for dedendum flank =[- 0.5,0]；λ=[0,0.5] is taken for addendum flank, then can obtain part helix face：

6th step：By the Mathematical Modeling obtained by previous step around 66.095 ° of X-axis rotate counterclockwise, then the involute circle for obtaining The intersection of post worm tooth-surface and horizontal plane is the Mathematical Modeling of its normal tooth profile：

7th step：The equation of involute FE on the left of being obtained in OXYZ coordinate systems：

8th step：Similarly, the equation to the 7th step involute FE carries out the conversion of the 4th step to the 6th step again, can this gradually The Mathematical Modeling of the left-hand screw face normal tooth profile that dextrorotation about the z axis of bursting at the seams is formed：

9th step：Design parameter and the 6th step according to involute cylindrical worm to be designed, the normal direction that the 8th step is released Three equations of the flank of tooth of tooth form, the intersection in the involute cylindrical worm flank of tooth He its normal section obtains intersection point, is gradually to open The normal tooth profile of line cylindrical worm；

Tenth step：The normal tooth profile of involute cylindrical worm is extracted, the parameter being related to：

The axial module of m-involute cylindrical worm to be extracted；

α -- the pressure angle of graduated circle of involute cylindrical worm to be extracted；

α_t-- the transverse pressure angle of involute cylindrical worm to be extracted；

γ -- the lead angle of involute cylindrical worm to be extracted；

γ_b-- the base lead angle of involute cylindrical worm to be extracted；

d₁-- the reference diameter of involute cylindrical worm to be extracted；

z₁-- the head number of involute cylindrical worm to be extracted；

The centre-to-centre spacing of a-- involute cylindrical worms to be extracted；

The facewidth of b-- involute cylindrical worms to be extracted；

The sequence number of the head number of n-- involute cylindrical worms to be extracted；

r_b-- the base radius of involute cylindrical worm to be extracted；

-- the root radius of involute cylindrical worm to be extracted；

-- the radius of addendum of involute cylindrical worm to be extracted；

The cross variable that the flank profil of μ-leading involute cylindrical worm is formed；

σ -- the half of reference circle central angle corresponding to transverse tooth thickness on the reference circle of involute cylindrical worm to be extracted；

The helix parameter of p-- involute cylindrical worms to be extracted；

The flank profil of λ-leading involute cylindrical worm to be extracted does spatially spiral and moves the cross variable to form tooth form；

The half of θ-central angle corresponding to outside circle camber line.

Using the present invention, the tooth form of involute cylindrical worm can be directly and accurately extracted, compared with prior art, no Only tooth form is accurate, and tooth form Mathematical Modeling is complete, extracts tooth form efficiency and greatly improves.

By foregoing invention, the tooth form extracting method of involute cylindrical worm had both been avoided and had used involute cylindrical worm End section involute approximately causes the tooth form extracted to produce larger error instead of the way of normal tooth profile, and accurate and complete Give the Mathematical Modeling and its figure of normal tooth profile.So that the accuracy and efficiency of tooth form extracting method are ensured.

Claims (1)

1. the tooth form extracting method of involute helicoid worm, it is characterised in that the extracting method comprises the following steps：

The first step：The parameter of known involute cylindrical worm to be designed is:Axial module m, head number z1, rotation direction, pressure angle α, in The heart is away from a, lead angle γ, reference diameter d₁；

Second step：It is XY faces with the plane vertical with the axis of involute cylindrical worm to be designed, with involute cylinder to be designed The axis of worm screw is Z axis, sets up OXYZ coordinate systems, wherein, tooth form is symmetrical on X-axis；Right side can be obtained in OXYZ coordinate systems gradually Burst at the seams the equation of HG：

$\left\{\begin{array}{c}{x}_u=r_b\·\cos (\mathrm{\μ}-\mathrm{\σ})+r_b\·\mathrm{\μ}\·\sin (\mathrm{\μ}-\mathrm{\σ})\\ y_u=-r_b\·\sin (\mathrm{\μ}-\mathrm{\σ})+r_b\·\mathrm{\μ}\·\cos (\mathrm{\μ}-\mathrm{\σ})\end{array}\right.$

In formula：x_u,y_uIt is the coordinate of any one point on involute HG；

The cross variable that the flank profil of μ-leading involute cylindrical worm is formed；σ -- on the reference circle of involute cylindrical worm to be extracted The half of reference circle central angle corresponding to transverse tooth thickness；r_b-- the base radius of involute cylindrical worm to be extracted；

3rd step：For bull involute cylindrical worm, tooth form right side involute HG need to be through such as down conversion

$\left\{\begin{array}{c}{x}_{11}=x_u\·\cos \[(n-1)\·\frac{360}{z_1}\]-y_u\·\sin \[(n-1)\·\frac{360}{z_1}\]\\ y_{11}=x_u\·\sin \[(n-1)\·\frac{360}{z_1}\]+y_u\·\cos \[(n-1)\·\frac{360}{z_1}\]\end{array}\right.$

In formula：x₁₁,y₁₁It is the coordinate of any one point on the upper right side involute HG of multistart worm n-th；

The sequence number of the head number of n-- involute cylindrical worms to be extracted；z₁-- the head number of involute cylindrical worm to be extracted；

4th step：Involute HG is spinned about the z axis to move and forms right side helicoid, formula, the involute are turned round according to vector The equation (left-handed z takes negative sign in equation, and dextrorotation takes positive sign) of helicoid：

$\left\{\begin{array}{c}x=x_{11}\·cos\mathrm{\λ}-y_{11}\·sin\mathrm{\λ}\\ y=x_{11}\·sin\mathrm{\λ}+y_{11}\·\cos \mathrm{\λ}\\ z=z_{11}\±p\·\mathrm{\λ}\end{array}\right.$

In formula：x₁₁,y₁₁, z₁₁It is the coordinate of any one point on the upper right side involute HG of multistart worm n-th；

X, y, z are the seats that n-th right side involute HG of multistart worm does any one point on the involute HG after spatial alternation Mark；

The flank profil of λ-leading involute cylindrical worm to be extracted does spatially spiral and moves the cross variable to form tooth form；P-- is to be extracted The helix parameter of involute cylindrical worm；

5th step：By positive and negative both direction rotation of the transverse profile in X-O-Y planes about the z axis, i.e. z₁₁=0；By λ according to setting Meter requires the neighbouring value 0, then obtain part helix face：

$\left\{\begin{array}{c}{x}_{111}=x_{11}\·cos\mathrm{\λ}-y_{11}\·sin\mathrm{\λ}\\ y_{111}=x_{11}\·sin\mathrm{\λ}+y_{11}\·\cos \mathrm{\λ}\\ z_{111}=\±p\·\mathrm{\λ}\end{array}\right.$

In formula：x₁₁₁, y₁₁₁, z₁₁₁It is that n-th right side involute HG of multistart worm does the λ=0 part tooth formed after spatial alternation The coordinate of any one point on the involute HG in face；

The flank profil of λ-leading involute cylindrical worm to be extracted does spatially spiral and moves the cross variable to form tooth form；P-- is to be extracted The helix parameter of involute cylindrical worm；

6th step：By the Mathematical Modeling obtained by step 5 around X-axis rotate counterclockwiseThe involute cylindrical worm for then obtaining The intersection of the flank of tooth and horizontal plane is the Mathematical Modeling of its normal tooth profile：

$\left\{\begin{array}{c}x=x_{111}\\ y=y_{111}\·cos(\frac{\mathrm{\π}}{2}-\mathrm{\γ})-z_{111}\·sin(\frac{\mathrm{\π}}{2}-\mathrm{\γ})\\ z=y_{111}\·\sin (\frac{\mathrm{\π}}{2}-\mathrm{\γ})+z_{111}\·\cos (\frac{\mathrm{\π}}{2}-\mathrm{\γ})\\ z=0\end{array}\right.$

Z=0-- horizontal planes, that is, the normal tooth profile chosen is the intersection of the involute helicoid worm flank of tooth and the plane where z=0

7th step：The equation of involute FE on the left of being obtained in OXYZ coordinate systems：

$\left\{\begin{array}{c}{x}_u=r_b\·cos(\mathrm{\μ}-\mathrm{\σ})+r_b\·\mathrm{\μ}\·sin(\mathrm{\μ}-\mathrm{\σ})\\ y_u=r_b\·sin(\mathrm{\μ}-\mathrm{\σ})-r_b\·\mathrm{\μ}\·cos(\mathrm{\μ}-\mathrm{\σ})\end{array}\right.$

8th step：Similarly, the equation to the 7th step involute FE carries out the conversion of the 4th step to the 6th step again, can the involute About the z axis dextrorotation formed left-hand screw face normal tooth profile Mathematical Modeling：

$\left\{\begin{array}{c}x=x_{222}\\ y=y_{222}\·cos(\frac{\mathrm{\π}}{2}-\mathrm{\γ})-z_{222}\·sin(\frac{\mathrm{\π}}{2}-\mathrm{\γ})\\ z=y_{222}\·cos(\frac{\mathrm{\π}}{2}-\mathrm{\γ})+z_{222}\·\cos (\frac{\mathrm{\π}}{2}-\mathrm{\γ})\\ z=0\end{array}\right.$

x₂₂₂, y₂₂₂, z₂₂₂It is that n-th right side involute FE of multistart worm does the λ=0 part flank of tooth formed after spatial alternation The coordinate of any one point on involute FE；

9th step：Design parameter and the 6th step according to involute cylindrical worm to be designed, the normal tooth profile that the 8th step is released Three equations of the flank of tooth, the intersection in the involute cylindrical worm flank of tooth He its normal section obtains intersection point, is involute circle The normal tooth profile of post worm screw；

Tenth step：The normal tooth profile of involute cylindrical worm is extracted, the parameter being related to：

The axial module of m-involute cylindrical worm to be extracted；

α -- the pressure angle of graduated circle of involute cylindrical worm to be extracted；

α_t-- the transverse pressure angle of involute cylindrical worm to be extracted；

γ -- the lead angle of involute cylindrical worm to be extracted；

γ_b-- the base lead angle of involute cylindrical worm to be extracted；

d₁-- the reference diameter of involute cylindrical worm to be extracted；

z₁-- the head number of involute cylindrical worm to be extracted；

The centre-to-centre spacing of a-- involute cylindrical worms to be extracted；

The facewidth of b-- involute cylindrical worms to be extracted；

The sequence number of the head number of n-- involute cylindrical worms to be extracted；

r_b-- the base radius of involute cylindrical worm to be extracted；

-- the root radius of involute cylindrical worm to be extracted；

-- the radius of addendum of involute cylindrical worm to be extracted；

The cross variable that the flank profil of μ-leading involute cylindrical worm is formed；

σ -- the half of reference circle central angle corresponding to transverse tooth thickness on the reference circle of involute cylindrical worm to be extracted；

The helix parameter of p-- involute cylindrical worms to be extracted；

The flank profil of λ-leading involute cylindrical worm to be extracted does spatially spiral and moves the cross variable to form tooth form；

The half of θ-central angle corresponding to outside circle camber line.